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Hezekiah Grayer
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Princeton University
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Analysis of fluids, fusion and space plasma
Analysis of fluids, fusion and space plasma
Advisor: Peter Constantin
Abstract:
An analysis of three distinct systems of partial differential equations is developed to rigorously constrain dynamical phenomena in fluids, fusion and space plasma.
Chapter 1. In the Earth’s mantle, natural convection may be modeled by the Stokes–transport equations. In unbounded two-dimensional space, it is an open question whether Stokes–transport is well-posed for bounded densities of finite mass. We establish the global-in-time well-posedness for bounded densities of compact support via an analysis of singular integrals. The long-time dynamics of density patches, idealizing plumes, is then studied. We prove the global-in-time persistence of higher Hölder regularity of the patch boundary via geometric analysis techniques.
Chapter 2. In tokamak and stellarator devices, plasma is controlled by a strong magnetic field with toroidal magnetic surfaces. Under hydrostatic equilibrium, heat in such strongly magnetized plasma is governed by an anisotropic diffusion equation. Diffusion is biased in the direction of the magnetic field and degenerates in the strong magnetization limit. We precisely characterize the limiting distribution of heat for toroidally fibered magnetic fields with ergodic field lines. Quantitatively, the temperature is well-approximated by a function that is constant on the invariant magnetic surfaces. Our result applies to several integrable and weakly nonintegrable examples in magnetic confinement fusion.
Chapter 3. In the vicinity of supernovae and other high-energy astrophysical settings, collisionless shocks in plasma are frequently observed. Also coincident are observations of significant radiation and relativistic particles whose energy distributions have nonthermal tails. Relativistic particles transfer kinetic energy into radiation upon acceleration, and this deceleration maybe modeled semiclassically by radiation reaction forces. The kinetic model of collisionless radiative plasma is given by the radiative Vlasov–Maxwell equations with certain radiation reaction forces. For a modified radiation reaction force, we demonstrate the global-in-time regularity of solutions for arbitrarily large initial data if the initial particle distribution has thermal tails.